using DifferentialEquations,WGLMakie, JSServe
#WGLMakie.activate!()
Page(exportable=true, offline=true)

Model 1

Construct and solve and ODE problem to calculate the change in X over time

function model1(du,u,p,t)
    M, K = p
    X,V = u
    du[1] = V
    du[2] = -K*X/M
    
end
M,K = 1.0,0.5
u0 = [0.5,1.0]
p = [M,K]
# construct an ODE problem
prob = ODEProblem(model1,u0,(1.0,20.0),p)
# solve using Tsitouras 5/4 Runge-Kutta method
sol = solve(prob, Tsit5(), saveat = 0.01);

Calculate the change in position and velocity for any given current position and velocity. Confusingly enough, $X$ is in the $y$ direction. As it's a spring, it feels intuitve to model its displacement vervically.

points = vec(Point.([-1:0.1:1]...,[-1:0.1:1]'...))
    f((x,y)) = (-K*y/M,x) ./ (sum(abs.((-K*y/M,x))) * 20)
    c((x,y)) = sum(abs.((-K*y/M,x)))

c (generic function with 1 method)

Put them together in a plot

fig = Figure(resolution = (1600,800))
    
    lines(fig[1,1],sol.t,hcat(sol.u...)[1,:], 
    axis = (
        xlabel = L"t",
        ylabel = L"X"
    ),
)
   arrows(fig[1,2],points,f.(points) , color = c.(points), 
    axis = (
        xlabel = L"\frac{dx^{2}}{dt^{2}}",
        ylabel = L"X"
    ),
)
fig[0, :] = Label(fig, L"\frac{dx^{2}}{dt^{2}} = -\frac{KX}{M}",
    textsize = 24)
fig

Model 2

Construct and solve and ODE problem to calculate the change in X over time

function model2(du,u,p,t)
    M, K, K₁ = p
    X,V = u
    du[1] = V
    du[2] = (-K*X- K₁*V)/M  
end
K₁ = 0.5
u0 = [0.5,1.0]
p = [M,K,K₁]
prob = ODEProblem(model2,u0,(1.0,20.0),p)
sol = solve(prob, Tsit5(), saveat = 0.01);

Calculate the change in position and velocity for any given current position and velocity

points = vec(Point.([-1:0.1:1]...,[-1:0.1:1]'...))
    k = 2
    m = 1
    f2((x,y)) = ((-K*y- K₁*x)/M,x) ./ (sum(abs.(((-K*y- K₁*x)/M,x))) * 20)
    c2((x,y)) = sum(abs.(((-K*y- K₁*x)/M,x)))

c2 (generic function with 1 method)

Put them together in a plot with a slider to control the $K_{1}$ value.

App() do session::Session
    K₁_slider = Slider(0.0:0.01:3.0)
    fig = Figure(resolution = (1600,800))
    t = sol.t
    u = map(K₁_slider) do val  
        Nprob = remake(prob, p = [M,K,val])
        sol = solve(Nprob, Tsit5(), saveat = 0.01)
        return hcat(sol.u...)[1,:]
    end

    dirs = map(K₁_slider) do val 
        K₁ = val 
        f2((x,y)) = ((-K*y- K₁*x)/M,x) ./ (sum(abs.(((-K*y- K₁*x)/M,x))) * 20)
        return f2.(points)
    end
    cols = map(K₁_slider) do val 
        K₁ = val 
        c2((x,y)) = sum(abs.(((-K*y- K₁*x)/M,x)))
        return c2.(points)
    end

    lines(fig[1,1],t,u, 
    axis = (
        xlabel = L"t",
        ylabel = L"X"
    ),
)
   arrows(fig[1,2],points,dirs , color = cols, 
    axis = (
        xlabel = L"\frac{dx^{2}}{dt^{2}}",
        ylabel = L"X"
    ),
)
fig[0, :] = Label(fig, L"\frac{dx^{2}}{dt^{2}} = \frac{-KX - K_{1}V}{M}",
    textsize = 24)
fig

    slider = DOM.div("K₁: ", K₁_slider, K₁_slider.value)
    return JSServe.record_states(session, DOM.div(slider, fig))
end

As you can see, setting the slider to zero recapture model 1.

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